A bag contains 6 marbles: 1 red, 2 blue, and 3 green; I reach into the bag and pick 4 marbles (simultaneously) at random. Determine the probability...

1. A bag contains 6 marbles: 1 red, 2 blue, and 3 green; I reach into the bag and pick 4 marbles (simultaneously) at random. Determine the probability that my 4-pick has ... (a) at least 2 green marbles. (b) all three colors represented. (c) at most two colors represented. 6 points Work your probabilities out completely, leaving them as fractions in their simplest form.

2. Of the 7-digit passwords formed by mixing 3 numbers (chosen from 1 to 9, without repeats) and 4 letters (chosen from A to Z, allowing repeats), I pick one at random. Determine the probability that my password ... (a) Has four A's. (b) Has at least two A's. (c) Begins and ends with an A. 7 points (d) Has two numbers next to each other with the third one not next to them. You may leave your probabilities in terms of mn, Pm n , Cm n , and factorials.

3. Of the 9-letter passwords formed by rearranging the letters AAAABBCCC, I select one at random. Determine the probability that my password ... (a) Is a palindrome. (b) Has A in the middle. (c) Begins and ends with B. 7 points (d) Has no two C's next to each other. You may leave your probabilities in terms of mn, Pm n , Cm n , and factorials.